$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 10.3.1.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{D}}= \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category, and let $Y$ be an object of $\operatorname{\mathcal{C}}$ (which we also regard as an object of $\operatorname{\mathcal{D}}$). Then the construction $(\operatorname{\mathcal{D}}^{0}_{/Y} \subseteq \operatorname{\mathcal{D}}_{/Y}) \mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}^{0}_{/Y} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}_{/Y}) } \operatorname{\mathcal{C}}_{/Y}$ induces a bijection
\[ \{ \textnormal{Sieves $\operatorname{\mathcal{D}}^{0}_{/Y} \subseteq \operatorname{\mathcal{D}}_{/Y}$} \} \xrightarrow {\sim } \{ \textnormal{Sieves $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$} \} . \]
Proof of Proposition 10.3.1.22.
Let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. We wish to show that there is a unique sieve $\operatorname{\mathcal{D}}^{0}_{/Y} \subseteq \operatorname{\mathcal{D}}_{/Y}$ with the following property: a morphism $f: X \rightarrow Y$ belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ if and only if the homotopy class $[f]$ belongs to the sieve $\operatorname{\mathcal{D}}^{0}_{/Y}$. The uniqueness assertion is immediate. To prove existence, we define $\operatorname{\mathcal{D}}^{0}_{/Y}$ to be the full subcategory of $\operatorname{\mathcal{D}}_{/Y}$ spanned by those homotopy classes $[f]: X \rightarrow Y$ such that $f$ belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$; by virtue of Remark 10.3.1.21, this condition depends only on the homotopy class $[f]$ and not on the choice of representative $f$. To complete the proof, it will suffice to show that the subcategory $\operatorname{\mathcal{D}}^{0}_{/Y} \subseteq \operatorname{\mathcal{D}}_{/Y}$ is a sieve. Suppose we are given a commutative diagram
10.20
\begin{equation} \begin{gathered}\label{equation:sieve-in-homotopy} \xymatrix@R =50pt@C=50pt{ X' \ar [dr]_{ [f'] } \ar [rr] & & X \ar [dl]^{ [f] } \\ & Y & } \end{gathered} \end{equation}
in the homotopy category $\operatorname{\mathcal{D}}= \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. We wish to show that if $[f]$ belongs to $\operatorname{\mathcal{D}}^{0}_{/Y}$, then $[f']$ also belongs to $\operatorname{\mathcal{D}}^{0}_{/Y}$. This follows from our assumption that $\operatorname{\mathcal{C}}^{0}_{/Y}$ is a sieve on $Y$, since (10.20) can be lifted to a $2$-simplex in the $\infty $-category $\operatorname{\mathcal{C}}$.
$\square$